Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 8 (1997), 114{119.
ON LOG-CONVEXITY OF A RATIO
OF GAMMA FUNCTIONS
Milan Merkle
We prove that the function x 7! (2x)=x 2(x) is strictly log-convex and the
function x 7! (2x)= 2(x) is strictly log-concave on x > 0 and we present some
consequences of these results.
1. INTRODUCTION
A positive function f is said to be logarithmically convex (or log-convex) if
the function x 7! log f(x) is convex. It is well known that the logarithmic convexity
is a fundamental property of the Gamma function [2, 3]. This paper continues our
study [7, 8, 9] of convexity and Schur-convexity of functions related to the Gamma
function. For a convenience, let us recall some basic facts about Schur-convexity.
More details can be found in [6].
Given two vectors x = (x1; x2; : : :; xn) and y = (y1 ; y2; : : :; yn ) of dimension
n, we say that x is majorized by y if
k
X
i=1
x[i] k
X
i=1
y[i] for k = 1; 2; : : :; n 1 and
n
X
i=1
xi =
n
X
i=1
yi ;
where (x[1] ; x[2]; : : :; x[n] ) is decreasing rearrangement of coordinates of x. If x is
majorized by y, we write x y. A function f of n variables is said to be Schurconvex on A Rn if
(1)
x y ) f(x) f(y)
for each x; y 2 A:
If x y implies f(x) < f(y) whenever x; y 2 A and x is not a permutation of y,
we say that f is a strictly Schur-convex function.
0
1991 Mathematics Subject Classication: 33A15, 26D20
114
On log-convexity of a ratio of gamma functions
Then
115
Let g be a continuous nonnegative function dened on an interval I R.
n
Y
f(x) = g(xi ); x 2 I n
i=1
is Schur-convex (strictly Schur-convex), on I n if and only if g is log-convex (strictly
log-convex) on I.
2. CONVEXITY RESULTS
Theorem 1. Dene
(2)
For x > 0, the
log-concave.
(2x)
F(x) = x (2x)
2(x) ; G(x) = 2 (x) :
function F is strictly log-convex and the
function
Proof. By the duplication formula for the Gamma function
G
(3)
(2x) = (2) 1=2 22x 1=2 (x) (x + 1=2)
(see [1] for example) we have that
F(x) = (2) 1=2 22x 1=2 (x(x++1=2)
1)
and therefore
+
1
X
1
1
00
(4)
(log F(x)) =
2 (x + 1)2 > 0 for x > 0:
k=0 (x + 1=2)
So, the function F is strictly log-convex on (0; +1).
The second derivative of the function x 7! logG(x) is
(5)
(log G(x))00 = 40 (2x) 20 (x);
where is the Digamma function and
+1
X
0 (x) = (x +1 k)2 :
k=0
By the duplication formula (3) we have
40 (2x) = 0 (x) + 0 (x + 1=2);
and we nd from (5) that
(log G(x))00 = 0 (x + 1=2) 0 (x) < 0; (x > 0):
is strictly
116
Milan Merkle
Hence, G is a strictly log-concave function on x > 0.
Corollary 1. For x > 0 and 2 [0; 1] we have the following inequalities:
2(2x + 1) (2x)
(2x
+
2)
x
+
(6)
2 (x + ) x
2 (x) ;
x+1
(7)
(2x + 2) 2(2x + 1) (2x) :
2(x + )
2(x)
x
The equality in both inequalities occurs if and only if = 0 or = 1. For
> 1, inequalities (6) and (7) hold with "" and "" interchanged.
Proof. The rst inequality follows from Jensen's inequality
(8)
log F(x + ) (1 ) log F(x) + F(x + 1)
for the log-convex function F dened in Theorem 1. The second inequality follows
from
(9)
log G(x + ) (1 ) log G(x) + G(x + 1);
where G is the log-concave function dened in Theorem 1. The statement about
equality is a consequence of the strict convexity of F and the strict concavity of G.
If > 1 then Jensen's inequalities (8) and (9) are reversed, and so (6) and (7) are
also reversed. 2
In a special case x = 1, Corollary 1 yields an interesting double inequality
+ ))
(10)
6 < (2(1
(0 < < 1);
2 (1 + ) < (1 + )3
and the reversed inequality for > 1.
Corollary 2.
The function
F1 (x) = log (2x + 1) 2 log (x + 1)
is strictly convex on x > 1=2.
Proof. It is easy to see that F1(x)00 = (log F(x))00, where F is dened in Theorem
1. Then from (4) we conclude that F1 is convex on x > 1=2.
Corollary 3. For all x 1 we have
(2x + 1) 2x ;
(11)
2(x + 1)
with equality if and only if x = 1. If 0 x < 1, then
(2x + 1) 2x ;
(12)
2(x + 1)
On log-convexity of a ratio of gamma functions
117
x = 0.
Proof. Inequality (11) is equivalent to
'(x) = log (2x + 1) 2 log (x + 1) x log 2 > 0:
For x 0 we have the following:
'0 (x) = 2 (2x + 1) 2 (x + 1) log 2; '00(x) = F100(x) > 0;
where is the Digamma function and F1 is the function dened in the statement of
Corollary 2. The recurence relation (z+1) = (z)+1=z yields '0(1) = 1 log2 > 0
and by '00(x) > 0 we have that '0(x) > 0 for x > 1; therefore '(x) > '(1) = 0.
On the other hand, since '(0) = 0, '0 (0) < 0 and '00(x) > 0, ' must be unimodal.
Then from '(1) = 0 it follows that '(x) < 0 for 0 < x < 1, which is equivalent to
(12).
with equality if and only if
Corollary 4.
The function
n (2x + 1)
Y
i
2 (xi + 1)
i=1
is strictly Schur-convex on x = (x1; : : :; xn) 2 ( 1=2; +1)n.
Proof. By Corollary 2, the function x 7! (2x+1)= 2 (x+1) is strictly log-convex
on x > 1=2 and therefore by a result mentioned in Section 1, the function is
Schur-convex.
(x) =
3. SOLUTION TO A PROBLEM POSED BY MITRINOVIC AND
PEC ARIC
The inequality (11) is not very sharp for large x. It can be improved in many
ways. In fact, it is not dicult to show that from
(a) (b)
aa bb
2 ((a + b)=2) ((a + b)=2)a+b
(see [4]) with a = 2x + 1 and b = 1 we obtain an inequality which is sharper than
(11) for x > 1 + " where " can be determined numerically. However, the results
of Section 2 might be interesting from another viewpoint. In [11] the following
problem was proposed: Let x1; x2; : : :; xr ; x be nonnegative real numbers such that
x1 + + xr = x. Prove (possibly under some additional conditions) that
r 2 (x + 1)
Y
i
(13)
21x :
(2x
+
1)
i
i=1
From (12) it follows that (13) is false if 0 < xi < 1 for all i = 1; 2; : : :r. On the
other hand, since
(x; x; ; x) (x1; x2; : : :; xr ) x = xr = x1 + x2 +r + xr ;
118
Milan Merkle
from the Schur-convexity result of Corollary 4 it follows that
r 2 (x + 1) 2(x + 1) r
Y
i
< (2x + 1)
(2x
i + 1)
i=1
for r 2 and if not all xi are equal. Hence, a sucient condition for (13) to hold is
2
(x + 1) r 1 ;
(2x + 1)
2rx
which, according to Corollary 1, holds if and only if x = 0 or x 1. Therefore,
the inequality (13) holds for nonnegative real numbers if x=r = 0 (i.e. xi = 0 for
all i) or x=r 1 and the reverse inequality holds if 0 < xi < 1 for all i. In the
intermediate case, i.e. when 0 < x=r < 1, but with some xi > 1, a computation
shows that neither (13) nor the reverse inequality is generally valid.
In connection with these results, let us note that if xi = mi are integers, then
the inequality (11) yields
n (2m )!
Y
i 2s ;
s = m1 + + mn ;
2
(m
!)
i
i=1
which is sharper than Khintchine's result in [5] or [10, p. 194]:
n (2m )!
Y
i 2s :
m
!
i
i=1
However, both results for integers follow from the inequality
(2m)! 2m ;
(m!)2
which can be proved by elementary means.
REFERENCES
1. M. Abramowitz, I.A. Stegun: A Handbook of Mathematical Functions. New York,
1965.
2. Emil Artin: The Gamma Function. Holt,Rinehart and Winston, New York 1964,
translation from the German original of 1931.
3. H. Bohr, J. Mollerup: Laerbog i matematisk Analyse, III. Kopenhagen 1922.
4. J. D. Keckic, P. M. Vasic: Some inequalities for the Gamma function. Publ. Inst.
Math. Beograd. N. Ser. 11(25) (1971), 107{114.
On log-convexity of a ratio of gamma functions
119
 dyadische Bruche. Math. Z. 18 (1923), 109{116.
5. A. Khintchine: Uber
6. A. Marshall, I. Olkin: Inequalities : Theory of Majorization and Its Applications.
Academic Press, New York, 1979.
7. M. Merkle: Logarithmic convexity and inequalities for the gamma function. J. Math.
Analysis Appl., 203 (1996), 369{380.
8. M. Merkle: Convexity, Schur-convexity and bounds for the Gamma function involving
the Digamma function. Rocky Mountain J. Math., to appear.
9. M. Merkle: Conditions for convexity of a derivative and some applications to the
Gamma function. Aequationes Math., to appear.
10. D. S. Mitrinovic: Analytic Inequalities. Berlin-Heidelberg-New York, 1970.
11. D. S. Mitrinovic, J. E. Pecaric: Problem 2.91. Univ. Beograd. Publ. Elektrotehn.
Fak. Ser. Mat. 2 (1992), 106.
University of Belgrade,
Faculty of Electrical Engineering,
P.O.Box 35-54, 11120 Belgrade,
Yugoslavia
(Received September 2, 1997)